Determine whether the quadratic function f(x) = x^2 - 4x - 5 has a maximum or a minimum value. Then, find the maximum or minimum value.
We know from algebra 2 that quadratic functions can be written in the form "ax^2 + bx + c." to figure out whether a quadratic function has a maximum or a minimum, we'll need to look at the sign of the "a" term. A negative "a" turns the parabola upside down so that it has a maximum value at the vertex. A positive "a" signifies a parabola with a minimum value at the vertex. In the function, we're looking that, a = 1, POSITIVE 1, so the parabola will have a minimum a its vertex. The reason the minimum is at the vertex is because the minimum is defined as the lowest point on the function's graph. For a quadratic function with a positive 'a' value, the lowest value will be the vertex. We already know that the x-value of the vertex is x = -b/2a. Plug in a = 1 and b = -4 to find this x-value: x = -(-4)/(2 x 1) = 2 The minimum occurs at x = 2. To find the y-value that corresponds to this x-value, simply plug x=2 into the function and solve for y: f(2) = (2)^2 - 4(2) - 5 = -9. So, the minimum value of the function is -9.
Without graphing the function, find where f(x) = 3x^4 - 4x^3 -12x^2 + 5 is increasing and where it is decreasing.
The Increasing/Decreasing Test states that: a. If f'(x) >0 on an interval, then f is increasing on that interval, and b. if f'(x) <0 on an interval, then f is decreasing on that interval. In other words, the function is increasing wherever its derivative is positive, and it's decreasing wherever it's derivative is negative. Does this make sense? Think about it. The derivative of a function at a point is the slope of the tangent line to that function at that point. An increasing function has a POSITIVE slope (aka a positive derivative) and a decreasing function has a NEGATIVE slope (aka a negative derivative). So for this problem, Step 1 would be to take the derivative of the function. f'(x) = 12x^3 - 12x^2 - 24x Now, how do we figure out where this derivative function is positive and where it's negative? Let's think about the graph of a parabola for a minute...a regular-old parabola that looks like the bottom half of a smiley face. Looking at it from left to right, the slope is negative until you reach the bottom of the parabola. At the bottom, the graph goes horizontal for a second, indicating a slope of ZERO. Then, the slope turns to positive for the rest of the graph. As is true for this parabola, any time the slope of a continuous and differentiable graph changes from negative to positive or from positive to negative, it has to have a slope of ZERO at that transition point. All that said, if we can find the points where the slope (derivative) is zero, then we can use that information to find the increasing and decreasing intervals that we're looking for. Let's do that by setting the derivative equation equal to zero. 0 = 12x^3 - 12x^2 - 24x 0 = 12x(x-2)(x+1) so x = -1, 0 and 2 These x values are called your critical points. To find the increasing/decreasing intervals, we will test points on either end of and in between these critical points to see if the derivative is positive or negative at these points That means we'll need to test a number that's less than -1, in between -1 and 0, in between 0 and 2, and greater than 2. Any numbers that meet these criteria will work. I'm going to use x = -2, x = -1/2, x = 1 and x = 3. I will plug these x values individually into the derivative equation to see if they come out positive or negative: f'(-2) = NEGATIVE f'(-1/2) = POSITIVE f'(1) = NEGATIVE f'(3) = POSITIVE Since NEGATIVE signifies decreasing and POSTIIVE signifies increasing, we can now say that: f(x) is decreasing on (-infinity, -1) and (0, 2) and f(x) is increasing on (-1,0) and (2, infinity). If you want to verify your answer, graph the original function, f(x), and look at where its slope is increasing and decreasing. The intervals you observe from the graph should match our answer!
Mary is walking to her friend's house at a rate of 3 miles per hour. Her friend is confused and thought that they were meeting at Mary's house, so she is walking towards Mary's house at a rate of 4 miles per hour. The houses are 1.49 miles apart, and Mary and her friend both left their houses at the same time. After how many minutes will they run into each other?
Let's write down everything we know. We'll use an "m" for "Mary," an "f" for "friend" and a "t" for the total. Rate_m = 3 miles/hour Rate_f = 4 miles/hour Distance_t = 1.49 miles/hour Since we are given information about distances and rates, it is very likely that we'll need to use the equation that relates them to this problem: Distance = Rate x Time Using this formula, let's set up some equations for Mary and her friend and plug in the known information: Distance_m = Rate_m x Time_m Distance_m = (3 miles/hour) x Time_m Distance_f = Rate_f x Time_f Distance_f = (4 miles/hour) x Time_f We don't know how far Mary and her friend each traveled, but we were told in the problem statement that their total distance apart is 1.49 miles, so when they run into each other, they will have run a total distance of 1.49 miles between the two of them. Here's what that looks like in an equation: Distance_t = Distance_m + Distance_f 1.49 = Distance_m + Distance_f Since we found earlier that Distance_m = 3 x Time_m and Distance_f = 4 x Time_f, we can rewrite the above equation like this: 1.49 = 3 x Time_m + 4 x Time_f So now it appears that we have two variables, Time_m and Time_f. However, if we think about what these variables represent, we'll quickly realize that they're actually one in the same. In the above equation, Time_m represents the time it takes for Mary to get to the meeting point and Time_f represents the time it takes for her friend to get to the meeting point. If they left their houses at the exact same time, then they will each have been walking for the same amount of time when they meet up. Therefore, Time_m = Time_f = Time. So, once again altering our equation, we now have: 1.49 = 3 x Time + 4 x Time And now we can combine like terms and solve for Time! 1.49 = 7 x Time Time = 0.213 hours Yep, this answer will come out in hours because our rates were in miles per HOUR. The last step is to convert hours to minutes by multiplying your answer by 60 (because there are 60 minutes in one hour). Time = 0.213 x 60 = 12.8 minutes!